Nonexistence of ultra-subharmonic periodic solutions for a class of nonautonomous dynamic system

In the study of the nonlinear dynamics, Melnikov functionis widely used as a criterion to check whether subharmonic orultra-subharmonic bifurcation even chaos will occur in a perturbed Hamiltonsystem. However, for the most cases, the classical Melnikov method canmerely show the existence of subharmonic periodic orbits. Such a result isattributed to that only first order approximation is adopted in theclassical Melnikov method. So higher-order Melnikov method is developed todetermine the existence of the ultra-subharmonic periodic solution. In thispaper, a class of non-autonomous differential dynamic system is studied. Itis proved that if there exists a periodic solution in such a system, thesolution can only be subharmonic, and the existence of ultra-subharmonicperiodic solution is impossible. Moreover, the nonexistence of R-typeultra-subharmonic periodic solution defined for a specified planar system isalso confirmed. As an application of above conclusions, some typicalexamples are investigated. The results demonstrate that second-orderMelnikov method used to justify the existence of ultra-subharmonic periodicorbits in a planar perturbation system may lead to a wrong conclusion. Asimple geometric explanation is also provided.